1. **State the problem:** We have two numbers, one is positive and is 3 times the other. If we add 2 to the larger number and 5 to the smaller number, then one becomes twice the other. We need to find the smaller number. 2. **Define variables:** Let the smaller number be $x$. Then the larger number is $3x$. 3. **Write the condition after adding:** Adding 2 to the larger number gives $3x + 2$, and adding 5 to the smaller number gives $x + 5$. 4. **Set up the equation:** One number becomes twice the other. This gives two possible equations: a) $3x + 2 = 2(x + 5)$ b) $x + 5 = 2(3x + 2)$ 5. **Solve equation (a):** $$3x + 2 = 2x + 10$$ Subtract $2x$ from both sides: $$3x - 2x + 2 = 10$$ $$x + 2 = 10$$ Subtract 2: $$x = 8$$ 6. **Check if $x=8$ is valid:** Smaller number $= 8$ (positive), larger $= 3 \times 8 = 24$. After adding: larger $= 24 + 2 = 26$, smaller $= 8 + 5 = 13$. Check if one is twice the other: $26 = 2 \times 13$ is true. 7. **Solve equation (b):** $$x + 5 = 2(3x + 2)$$ $$x + 5 = 6x + 4$$ Subtract $x$ from both sides: $$5 = 5x + 4$$ Subtract 4: $$1 = 5x$$ $$x = \frac{1}{5}$$ 8. **Check if $x=\frac{1}{5}$ is valid:** Smaller number $= \frac{1}{5}$ (positive), larger $= 3 \times \frac{1}{5} = \frac{3}{5}$. After adding: larger $= \frac{3}{5} + 2 = \frac{3}{5} + \frac{10}{5} = \frac{13}{5}$, smaller $= \frac{1}{5} + 5 = \frac{1}{5} + \frac{25}{5} = \frac{26}{5}$. Check if one is twice the other: $\frac{26}{5} = 2 \times \frac{13}{5}$ is true. 9. **Conclusion:** Both $x=8$ and $x=\frac{1}{5}$ satisfy the conditions, but since the problem states "a positive number is 3 times another number," both are positive. The smaller number can be either $8$ or $\frac{1}{5}$ depending on which number is considered smaller initially. Since $x$ is defined as the smaller number, the smaller number is either $8$ or $\frac{1}{5}$. However, $8$ is larger than $\frac{1}{5}$, so the smaller number is $\frac{1}{5}$. **Final answer:** The smaller number is $\boxed{\frac{1}{5}}$.